Energies 2009, 2, 427-444; doi:10.3390/en20200427
OPEN ACCESS
energies
ISSN 1996-1073
www.mdpi.com/journal/energies
Article
A Microscale Modeling Tool for the Design and Optimization of
Solid Oxide Fuel Cells
Shixue Liu 1,2,3,4, Wei Kong 2 and Zijing Lin 1,2,*
1
2
3
4
Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and
Technology of China, Hefei 230026, China; E-Mail: sxliu@mail.ustc.edu.cn (S.L.)
Department of Physics, University of Science and Technology of China, Hefei 230026, China;
E-Mail: wkong@mail.ustc.edu.cn (W.K.)
Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Sciences,
Qingdao 266101, China
Key Laboratory of Biofuels, Chinese Academy of Sciences, Qingdao 266101, China
* Author to whom correspondence should be addressed; E-Mail: zjlin@ustc.edu.cn; Tel.: +86-5513606345; Fax: +86-551-3606348
Received: 22 May 2009; in revised form: 9 June 2009 / Accepted: 10 June 2009 /
Published: 23 June 2009
Abstract: A two dimensional numerical model of a solid oxide fuel cell (SOFC) with
electrode functional layers is presented. The model incorporates the partial differential
equations for mass transport, electric conduction and electrochemical reactions in the
electrode functional layers, the anode support layer, the cathode current collection layer
and at the electrode/electrolyte interfaces. A dusty gas model is used in modeling the gas
transport in porous electrodes. The model is capable of providing results in good
agreement with the experimental I-V relationship. Numerical examples are presented to
illustrate the applications of this numerical model as a tool for the design and optimization
of SOFCs. For a stack assembly of a pitch width of 2 mm and an interconnect-electrode
contact resistance of 0.025 Ωcm2, a typical SOFC stack cell should consist of a rib width of
0.9 mm, a cathode current collection layer thickness of 200–300 μm, a cathode functional
layer thickness of 20–40 μm, and an anode functional layer thickness of 10–20 μm in order
to achieve optimal performance.
Energies 2009, 2
428
Keywords: solid oxide fuel cell; functional layer; modeling tool; optimization; finite
element method
1. Introduction
Solid oxide fuel cells (SOFCs) are clean and efficient chemical to electrical energy transfer devices.
A traditional SOFC includes three layers: porous anode, dense electrolyte and porous cathode [1,2].
The fuel and air are transported from the gas channels to the three phase boundary (TPB) near the
electrode/electrolyte interfaces through the porous electrodes and the electrochemical reactions take
place in the TPBs. To improve the fuel cell performance by increasing the TPB length to reduce the
activation polarization while maintaining the concentration polarization at an acceptable level [3-7],
the new design of an anode-supported SOFC cell often consists of five layers: an anode support layer
(ASL), an anode functional layer (AFL), an electrolyte layer, a cathode functional layer (CFL) and a
cathode current collector layer (CCCL), as shown schematically in Figure 1. The thin functional
layers, CFL and AFL, are designed to increase the TPB length with low porosity and small particle
sizes. The thick layers, CCCL and ASL, are of high porosity and large particle sizes to enable easy gas
transport to the functional layers.
Figure 1. Schematic diagram of an anode-supported planar SOFC stack cell unit.
In order to achieve high performance for the fuel cells, the cell geometries should be carefully
optimized. The thickness of CFL and CCCL are usually the focus of structural optimization in
experimental and theoretical works due to their strong influences on the cathode activation polarization
and concentration polarization. The thickness of ASL is also an important parameter for the
performance optimization [8]. Experimentally, Zhao et al. have carried out a systematic study on the
influence of the microstructures of cell components on the cell properties [3] and provided the
following optimized cell parameters: an ASL thickness of 0.5 mm and porosity of 57%, an AFL
thickness of 20 μm, a CFL thickness of 20 μm and a CCCL thickness of 50 μm. The experiments by
Haanappel et al. [4] showed that a CFL thickness of at least 10 μm and a CCCL thickness of at least
45–50 μm were required for the optimal cell performance. As experiments are expensive and time
consuming, however, it is difficult for the experiments to explore a large number of design parameters
Energies 2009, 2
429
and different material choices. Numerical models that incorporate the proper physical mechanisms are
more versatile and flexible and are highly desirable for the guiding analyses of a large
engineering community.
Most existing theoretical and modeling works are concerned with the traditional SOFCs without the
functional layers [8-16]. For SOFCs with the functional layers, the electrochemical active areas are
extended from the electrode/electrolyte interfaces to the functional layers and gaseous species are
consumed and produced within the functional layers instead of only at the interfaces. The coupling of
the governing equations is not only by the boundary conditions but also by the source or sink terms
and the multi-physics theoretical models are complicated. Consequently, only recently have there been
a few theoretical models considering SOFCs with the functional layers [17-20].
In the present work, a two dimensional stack cell model is developed for SOFCs with the electrode
functional layers. Electrochemical reactions in the functional layers and on the electrode/electrolyte
interfaces are considered in detail. A dusty gas model is used to describe the gas transfer process in
porous layers and electric conducting equations are used to describe electron and oxygen ion
conducting processes. In fitting the experimental data for model parameters, the effects of the
interconnect ribs and the interconnect-electrode contact resistance on the fuel cell performance are also
considered. The model is then used to optimize several geometry parameters for stack cell.
2. Method
The 2D cross-section geometry model for the SOFC stack is shown in Figure 2. Here dchannel and drib
are one half of the channel width and one half of the interconnect rib width, respectively. The typical
geometric, material and operational parameters are shown in Table 1. The air and the fuel are supplied
to the reaction sites in the functional layers (CFL and AFL) from the gas channels through the outside
porous layers (CCCL and ASL). Two important electrochemical reactions occur in the functional
layers when cell operates with hydrogen fuel and air oxidant as illustrated in Figure 2. At the CFL
reaction site, each half of an oxygen molecule gets two electrons conducted from the nearest cathodic
interconnect rib and is reduced to an oxygen ion. The oxygen ion is then transported from the CFL
reaction site to the AFL reaction site through the electrolyte (YSZ). Finally at the AFL reaction site,
the oxygen ion reacts with a hydrogen molecule and produces a water molecule and two electrons and
the electrons are conducted to the nearest anodic interconnect rib.
The overall cell performance depends on the operating cell potential and the output current. The
operating cell potential (Vcell) can be formally expressed as [8]:
Vcell  E0 - ASR;a -conc;a -act;a - ohm;e;a -ohm;i;a - ohm;el - ohm;i;c -ohm;e;c - act;c - conc;c - ASR;c
(1)
where E0 is the Nernst potential, ηconc;a and ηconc;c are the concentration overpotentials in the anode and
cathode, respectively, ηact;a and ηact;c the anode and cathode activation overpotentials, ηohm;e;a, ηohm;e;c
the electronic ohmic overpotentials in the anode and cathode, ηohm;i;a, ηohm;el and ηohm;i;c the ionic ohmic
overpotentials in the anode, electrolyte and cathode, ηASR;a and ηASR;c the anode-rib and cathode-rib
interface overpotentials due to the contact resistance at the material boundaries.
Energies 2009, 2
430
Figure 2. Schematic cross-section model of half repeating unit of SOFC stack.
Combining the expressions for the Nernst potential and the concentration overpotentials,
pH ,f
pO ,air
pH ,f pH2O,AFL
pO ,air
RT
RT
RT
RT
E0  E00 
ln( 2 ) 
ln( 2 ) , conc;a 
ln( 2
) and conc;c 
ln( 2 ) ,
2F
pH2O,f
4F
1atm
2F
pH2O,f pH2 ,AFL
4F
po2 ,CFL
Equation 1 may be rewritten for any electrochemically active sites in AFL and CFL as
Vcell  E00 - ASR;a -a0 - act;a -ohm;e;a - ohm;i;a - ohm;el - ohm;i;c -ohm;e;c - act;c -c0 - ASR;c
(1a)
where subscripts f, air, AFL and CFL refer to the values in fuel channel, air channel, AFL area and
CFL area, respectively. E00 is the Nernst potential when the partial pressure of H2, H2O and O2 are all
at 1 atm ( E00 = 1.01 V at 973 K or 700 °C), R the universal gas constant, T the temperature at Kelvin,
F the Faraday constant. pH2 ,AFL , pH2O,AFL and pO2 ,CFL are the partial pressure of H2 and H2O at the
electrochemically active site in the AFL and the partial pressure of O2 at the active site in the CFL,
respectively.  a0 and  c0 are respectively the anode and cathode concentration balance potentials and
are calculated as:
a0  
pH ,AFL
RT
ln( 2
)
2F
pH2O,AFL
(2a)
po ,CFL
RT
ln( 2 )
4F
1atm
(2b)
c0  
The details of computing the other overpotential terms in Equation 1 will be described later. Notice,
however, the concentration balance overpotentials (a0 ,c0 ), the activation overpotentials ( act;a ,  act;c )
and the electrode ohmic overpotentials (ohm;e;a ,  ohm;i;a , ohm;i;c ,  ohm;e;c ) in Equation 1a and Equation 2
are dependent on the given active sites in the AFL and CFL layers that are fined meshed to reveal their
spatial variations.
The average output current density of the SOFC stack can be calculated by
j
1
d pitch
d pitch

x 0
jy dx
(3)
Energies 2009, 2
431
where dpitch is the pitch width and dpitch = dchannel + drib. The x and y axis are along the horizontal and
vertical directions in the 2D model, respectively. jy is the y component of the current density flux
vector.
Table 1. Geometric, material and operational parameters for SOFC with functional layers.
Parameters
Cell temperature, T (°C)
Inlet fuel/air pressure, Patm (Pa)
Cell output voltage, Vop (V)
ASL thickness, lASL (μm)
AFL thickness, lAFL (μm)
Electrolyte thickness, lYSZ (μm)
CFL thickness, lCSL (μm)
CCCL thickness, lCCCL (μm)
Porosity, 
Ni volume fraction,  Ni
LSM volume fraction,  LSM
Tortuosity, 
Angle of particle contact,  (o)
Bruggeman factor, m
Mean particle diameter, dp (μm)
Electrical conductivity of Ni,  Ni (s m-1)
Electrical conductivity of LSM,  LSM (s m-1)
Ionic conductivity of YSZ,  YSZ (s m-1)
Diffusion volume of H2, vH2 (m3 mol-1)
Diffusion volume of H2O, vH2O (m3 mol-1)
Diffusion volume of O2, vO2 (m3 mol-1)
Diffusion volume of N2, vN2 (m3 mol-1)
Molar mass of H2, M H 2 (kg mol-1)
Molar mass of H2O, M H 2O (kg mol-1)
Molar mass of O2 , M O 2 (kg mol-1)
Molar mass of N2 , M N 2 (kg mol-1)
Permeability of anode (m2)
Permeability of cathode (m2)
Viscosity of fuel (Pa s)
Viscosity of air (Pa s)
Knudsen diffusion coefficient of H2 (m2 s-1)
Knudsen diffusion coefficient of H2O (m2 s-1)
Knudsen diffusion coefficient of O2 (m2 s-1)
Knudsen diffusion coefficient of N2 (m2 s-1)
Value
700
1.013 × 105
0.7
1,000
20
8
20
50
0.48 (ASL), 0.23 (AFL),
0.26 (CFL), 0.45 (CCCL)
0.55 (ASL, AFL)
0.475 (CFL), 1 (CCCL)
3
30
1.5
1 (ASL, CCCL), 0.5 (AFL, CFL)
3.27 106  1065.3T
8.855 107 / T *exp(1082.5 / T )
3.34  10 4 exp(10300 / T )
7.07 × 10-6
12.7 × 10-6
16.6 × 10-6
17.9 × 10-6
2 × 10-3
18 × 10-3
32 × 10-3
28 × 10-3
7.93 × 10-16
3.06 × 10-16
2.8 × 10-5
4 × 10-5
4.37 × 10-4
1.46 × 10-4
7.64 × 10-5
8.17 × 10-5
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432
2.1. Electrochemical Reactions in AFL and CFL
In the present model, empirical cathodic (anodic) Butler–Volmer equations are used to describe the
electrochemical reaction rate functions in CFL (AFL) and on the CFL/electrolyte (AFL/electrolyte)
interface. The transfer current densities per TPB length (A/m1) in the AFL ( iTPB,a ) and CFL ( iTPB,c ) are
expressed [17] as:
iTPB,a
act ,a

pH2O  14, 000Pa

4 act ,a
0.11 0.67
pH2 pH 2O exp(10212 / T )
1.645  10

 act ,a

p  14, 000Pa
4


0.11

1.645 10 act ,a p 14, 0000.67 exp(10212 / T ) H 2O
H2

iTPB ,c 
RT exp(2 F act ,c / RT )  exp(2 F act ,c / RT )
4F
0.00136 pO0.25
exp(17401/ T )
2
(4a)
(4b)
where η is the activation polarization, pi the partial pressure of species i.
For the reaction occurs in the functional layers, the current source (A/m3) can be calculated as
V
V
iTPBTPB
. Here TPB
is the volume specific TPB length (mm-3) and can be expressed [17,21]
V
TPB
 6 d p nt  el  io Pel Pio sin

(5a)
2
where dp is the particle diameter, nt (  6(1   ) /  d 3p ) the number density of all particles,  j the
volume fraction of particle type j (el: electron conducting particle, io: ionic conducting particle or

YSZ), Pj (= (1  ((4.236  6 j ) / 2.472) 2.5

0.4
) the probability of particle type j to form percolated or
globally continuous clusters,  the angle of particle contact.
For the reaction occurs on the electrode/electrolyte interface, the current flow (A/m2) can be
A
A
. Here TPB
is the area specific TPB length (mm-2) and may be expressed [17]
calculated as iTPBTPB
A
TPB
  d 2 nt  el Pel sin

p
2
(5b)
The activation polarizations, ηact,a and ηact,c, are related to the electric and balance potentials by [8]
act,a  Ve,AFL  Vi,AFL  a0
(6a)
act,c  Vi,CFL  Ve,CFL  
(6b)
0
c
where Ve,AFL and Vi,AFL are respectively the electronic and ionic potentials at the electrochemically
active site in the AFL, Vi,CFL and Ve,CFL the ionic and electronic potentials at the electrochemically
active site in the CFL.
Energies 2009, 2
433
2.2. Gas Transport in Porous Electrode
2.2.1. Dusty gas model
The dusty gas model is used here as it is more accurate than the Darcy’s law and Maxwell–Stefen
model to describe the mass transport in porous electrodes. The original dusty gas model can be
expressed as [10,14,22,23]
n x N x N
Ni
1
j i
i
j



eff
eff
DiK
Dij
RT
j 1

kp 
 pxi  xi p  xi p eff 
DiK  

(7)
where Ni is the total molar flux of species i, xi (  ci /  c j ) the molar fraction, k the permeability, µ
j
eff
iK
the viscosity, p the total gas pressure, D
and
eff
ij
D
(= 3.198 108 
 Dij / 
(=
T
p (v  v )
1/3
i
)
1.75
1/3 2
j
[
the
(=  DiK /  ) the effective Knudsen diffusion coefficients
effective
binary
diffusion
coefficients,
Dij
1
1 1/ 2
] ) the binary diffusion coefficient, ε and τ the porosity and

Mi M j
tortuosity, ci, νi and Mi the molar concentration, diffusion volume and molar mass of species i,
respectively [24,25]. The required parameters are shown in Table 1.
For binary gas, Equation 7 may be reduced to
N i   Di ci  ci u
(8)
where the effective diffusion coefficients of the species, Di, are given by
D1 
D2 
D12eff
D12eff D1effK
 x1 D2effK  x2 D1effK
(9a)
D12eff
D12eff D2effK
 x1 D2effK  x2 D1effK
(9b)
The effective molar flow velocity u is given by

D1effK D2effK
k
 p

u  
 RTctot  D12eff  x1 D2effK  x2 D1effK   


(10)
2.2.2. Governing equations
The mass transport processes in porous electrodes are governed by the mass diffusion and
convective equations. For species i, the transport equation can be expressed by [26]
  N i  Ri
(11)
where Ri is the reaction rate of species i (mol m-3 s-1) in the functional layers and Ri  0 in ASL and
V
CCCL because there is no reaction, RH2  iTPB,a TPB
/ 2 F for the hydrogen consumption and
V
V
RH2O  iTPB,a TPB
/ 2 F for the water steam production in AFL and RO2  iTPB,c TPB
/ 4 F for the oxygen
consumption in CFL.
Energies 2009, 2
434
2.3. Electrical Conduction
The electronic charge transfer in the electrodes (ASL, AFL, CFL and CCCL) are governed by:
  ( eeff Ve )  Qe
(12a)
where  eeff is the effective electronic conductivity of the electrode, Ve the electric potential in the
electrode.  eeff Ve is the flux vector of the electronic current density. Qe is the electronic current
source (A/m3) and Qe  0 in ASL and CCCL because there is no electrochemical reaction,
V
V
in AFL and Qe  iTPB,c TPB,c
in CFL. The electronic potential differences along the
Qe  iTPB,a TPB,a
electrical current flux paths yield the ohmic overpotentials, ηohm;e;a and ηohm;e;c.
The ionic charge transfer in the functional layers (AFL, CFL) and electrolyte are governed by
  ( ieff Vi )  Qi
(12b)
where  ieff is the effective ionic conductivity of the functional layers or the electrolyte, Vi the electric
potential.  ieff Vi is the flux vector of the ionic current density. Qi is the oxygen ionic current source
V
in AFL
(A/m3) and Qi  0 in ASL and CCCL as there is no electrochemical reaction, Qi  iTPB,a TPB,a
V
in CFL. The ionic potential differences along the ionic current flux paths yield
and Qi  iTPB,c TPB,c
the ohmic overpotentials, ηohm;i;a, ηohm;i;c and ηohm;el.
The effective conductivity can be expressed [17]:
m
 eff
j   j ((1   ) j Pj )
(13)
where σj is the conductivity of species j, m Bruggeman factor considering the effects of tortuous
conduction paths and constricted contact areas between particles.
The electric potential loss inside the interconnect plate is assumed to be negligible due to the high
conductivity of the metallic material. The local current densities cross the interconnect/anode ( jIa )
and the cathode/interconnect ( jcI ) interfaces are determined by the associated electric potential
changes, or the interface overpotentials:
jIa 
jcI
Ve,I/a  Ve,a/I

 ASR;a
ASRcontact
ASRcontact
V V
 ASR;c
 e,c/I e,I/c 
ASRcontact
ASRcontact
(14a)
(14b)
where Ve,I/a and Ve,a/I are respectively the interconnect and ASL electric potentials at the anodeinterconnect boundary, Ve,c/I and Ve,I/c the CCCL and interconnect electric potentials at the cathodeinterconnect boundary, ASRcontact the area specific contact resistance.
2.4. Boundary Conditions (BCs)
The boundary settings for the mass transport equations are shown in Table 2. The molar
concentrations at the channel/electrode interface are related to the molar fractions by the ideal gas
equation of state, and can be calculated by ci0  xi0 Patm / RT .
Energies 2009, 2
435
The boundary settings for the electronic and ionic charge transfer equations are shown in Table 3.
For the electronic charge transfer equation the current flow on the electrode/Electrolyte interface is
inward/outward current boundary, while for the ionic charge transfer equation the current flow is
interior current source. The contact resistance is set on the interface between interconnect ribs and
the electrodes.
Table 2. Boundary settings for mass transports in electrodes.
Equations
Boundary
Fuel
BC Type
transfer
ASL/channel interface
H2O molar
H2 inward molar
H2O inward
concentration
concentration
flux
molar flux
c 0H
c 0H O
A
iTPB,a TPB
/ 2F
A
iTPB,a TPB
/ 2F
BC
BC Type
transfer
All others
H2 molar
Boundary
Air
AFL/Electrolyte interface
2
2
CCCL/channel interface
CFL/Electrolyte interface
All others
O2 molar
N2 molar
O2 inward molar
N2 inward
concentration
concentration
flux
molar flux
c O0
c 0N
A
iTPB,c TPB
/ 4F
0
BC
2
2
Insulation/Symmetry
Insulation/Symmetry
Table 3. Boundary settings for the electronic and ionic charge transfer equations.
Equations
Boundary
Electronic
BC Type
transfer
BC
Ionic
transfer
BC Type
BC
Rib/CCCL
Rib/ASL
CFL/Electrolyt
AFL/Electrolyt
interface
interface
e interface
e interface
Reference
Reference
Inward current
Inward current
Electric
potential
potential
flow
flow
insulation
0
A
iTPB,c TPB
A
iTPB,a TPB
Interior current
Interior current
Electric
source
source
insulation
0
Vcell- E0
i

A
TPB,c TPB
All others
iTPB,a 
A
TPB
2.5. Numerical Method
The finite element commercial software COMSOL MULTIPHSICS® Version 3.4 [26] was used in
the present study to solve the required partial differential equations (PDEs) such as the mass diffusion
and convective equations, the electronic and ionic charge transfer equations. The Butler–Volmer
equations are integrated into the PDEs as sources or boundary settings. The COMSOL stationary
nonlinear solver uses an affine invariant form of the damped Newton method [26] to solve the
discretized PDEs with a relative tolerance of 1 × 10-6. Free triangle meshes with a maximum element
size of 5 × 10-6 m for electrolyte and AFL areas were used.
It is worth mentioning a subtle feature in the numerical solution process. According to Equation 4b,
some current may be generated by the electrochemical reaction even when PO2 is very small and may
result in some small negative value for PO2 due to the oxygen consumption by the current production.
This in turn may cause numerical failure for Equation 4b. To overcome the numerical problem, oxygen
Energies 2009, 2
436
molar fraction was set as xO2  max( xO2 ,106 ) . Similarly, the activation polarizations were set to be
non-negative,act,a  max(act,a , 0) and act,c  max(act,c , 0) , to comply with the physical requirement.
3. Results and Discussion
3.1. Model Fitting of the Experimental I-V Curve
As shown in [3], silver meshes were pressed against CCCL and ASL for single cell testing. The
meshes may be regarded as interconnect ribs with small pitch width. Here a pitch width of 0.3 mm and
a rib width of 0.05 mm corresponding to the experimental description were used for the model fitting.
When the contact resistance is 0.008 Ωcm2 and the boundary hydrogen/oxygen molar fraction is
0.97/0.21, the theoretical curve shown in Figure 3 agrees well with the experimental data. It should be
noted here that when the pitch width is small, the current through the electrode/rib interface jIa and
jcI are uniform [8], then the effective contact resistance is approximately
eff
ASRcontact
 2 ASRcontact d pitch / d rib . The effective contact resistance is an effective value assuming that the
contact resistance was distributed over the entire pitch width instead of the rib width. Accordingly,
eff
is found here to be about 0.096 Ωcm2, in good agreement with the experimental estimation
ASRcontact
that the total ohmic loss is 0.19 Ωcm2 at 700 °C and half the total ohmic loss is due to the contact
resistance [3]. The agreement between the theoretical and experimental results validates the numerical
model described here. Moreover, it is striking to see that the interconnect-electrode contact resistance
is magnified by a factor of 2dpitch/drib and even a very small contact resistance may substantially affect
the fuel cell testing result.
Figure 3. Comparison of theoretical and experimental I-V curves.
The ionic current density distribution is quite uniform along the x direction and may be viewed as
one-dimensional along the y direction in the model for the testing single cell. Figure 4 shows the ionic
current density in the CFL, electrolyte and AFL at x = 0 when operating cell potential is 0.7 V. The
maximum current density across the electrolyte is 0.64 A/cm2. The current densities generated by the
functional layer body reaction and the functional layer/electrolyte boundary reaction are respectively
0.591 A/cm2 and 0.049 A/cm2 in the cathode side, and 0.439 A/cm2 and 0.201 A/cm2 in the anode side.
Energies 2009, 2
437
This means that the body reaction is the dominant contributor for the fuel cell current generation and
the model picture with only boundary reactions may be too simplified.
Figure 4. The ionic current density distribution in the CFL, electrolyte and AFL at x = 0.
As shown in Figure 4, the ionic current density in AFL decreases rapidly with the distance from the
AFL/electrolyte interface, and almost all current is generated in the 4 μm region around the interface.
The electrochemical reaction occurs in the whole area of CFL, but a layer thickness of 11 μm near the
interface contributes about 90% of the total current density. This indicates that the effective layer
thicknesses for the AFL and CFL electrochemistry reactions are quite small. As the cathode activation
overpotential for a given current density is higher than the anode counterpart, the cathode side requires
an effective area larger than the anode side in order to generate the same amount of the total current.
Therefore, the thickness of the effective cathode active layer is higher than that of the anode side.
3.2. Distributions of Physical Quantities in Stack Cell
The output current density is affected by the cathode layer thickness and a thicker layer benefits to
the oxygen transport and the stack performance [8]. Two CCCL thickness lCCCL, 50 μm and 200 μm,
are used to show the physical quantities in stack cell. The pitch width is 2 mm and the rib width is
0.8 mm. The boundary hydrogen and oxygen molar fraction are 0.7 and 0.21 respectively. The area
specific contact resistance on the anode/interconnect and the cathode/interconnect boundaries are
uniform (0.025 Ωcm2). Figure 5 shows the physical quantity distributions for lCCCL of 50 μm. The
average output current density is 0.374 A/cm2. Figure 6 shows the distributions for lCCCL of 200 μm
and the average output current density is 0.409 A/cm2.
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Figure 5. The distributions of physical quantities when the CCCL layer thickness is
50 μm: (a) hydrogen and oxygen molar fractions; (b) total electronic current density and
current flux vector; (c) the y component of the current flux vector.
(a)
(b)
(c)
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Figure 6. The distributions of physical quantities when the CCCL layer thickness is
200 μm: (a) hydrogen and oxygen molar fractions; (b) total current density and current flux
vector; (c) the y component of the current flux vector.
(a)
(b)
(c)
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440
The hydrogen and oxygen distributions for a CCCL layer of 50 μm are shown in Figure 5a and are
similar to the distributions in a stack cell without the functional layers [8]. The oxygen distribution in
the area under the rib is easily exhausted and the area is hardly active. With a thicker CCCL layer of
200 μm (Figure 6a), the oxygen transfer is more effective and can supply oxygen to the area under the
rib, leading to less concentration polarization and more active area. The existence of thin functional
layers almost has no effect on the gas transfer.
The total electronic current densities and the current flux vectors are shown Figures 5b and 6b. The
total current density in the CFL is obviously less than that in the CCCL. The reason is that the
electronic conductivity of CCCL (2.99 × 104 s m-1) is much largely than that of CFL (5307 s m-1), and
the current flux in CCCL has a much larger x-component than that in CFL to carry the current to the
interconnect rib. Understandably, the current density distribution is more uniform and with a less
ohmic polarization for a lCCCL of 200 μm than that for a lCCCL of 50 μm.
Figures 5c and 6c show the y component of the electronic current density flux. The values for AFL
and CFL other than at the AFL/electrolyte or CFL/electrolyte interfaces are approximately the local
current densities produced by the bulk and boundary electrochemical reactions. At the ASL/AFL
interface, the value decreases substantially from the area under gas channel to the area under the rib for
lCCCL = 50 μm (Figure 5c) while it is quite uniform for lCCCL = 200 μm (Figure 6c), also indicating that
the latter is more favorable.
3.3. Optimization of Geometry Parameters
3.3.1. Optimization of the CCCL thickness
As described in Section 3.2, different CCCL layer thicknesses for otherwise identical cells may
have different performances. Figure 7 shows the variations of the average output current density with
lCCCL. The current density increases by 9.4% when lCCCL increases from 50 to 200 μm, but the
difference is less than 0.4% when the lCCCL changes between 200 and 400 μm. Considering the
performance benefit and material cost, the optimized CCCL layer thickness should be 200–300 μm.
Unless stated otherwise, the layer thickness of 200 μm is used for optimizing other parameters
described below.
Figure 7. Variation of the average output current density with the CCCL layer thickness.
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441
3.3.2. Optimization of the rib width
The optimal rib width is the result of balancing the polarizations of the current collection and the
gas transport. The most important factor affecting the optimized rib width is the contact resistance [8].
Figure 8 shows the optimized rib width and the optimal output current density versus the area specific
contact resistance. Larger contact resistance requires larger optimized rib width to balance the
increased ohmic loss, ηASR;a and ηASR;c. The optimal current density decreases rapidly with the increase
of the contact resistance for two main reasons: a larger contact resistance leads to a higher ohmic loss
and the increased optimal rib width leads to a smaller active area. For the reference contact resistance
of 0.025 Ωcm2, the optimized rib width is 0.9 mm.
Figure 8. The optimized rib width and the optimal output current density vs the area
specific contact resistance.
3.3.3. Optimization of the CFL thickness
Limited by the catalytic ability of the cathode materials, the cathode activation polarization is often
the most important component of the total fuel cell polarization and a CFL layer with small particle
size and small porosity is desirable for enlarging the TPB length and reducing the activation
polarization. For a given microstructure, the CFL thickness is also an important factor affecting the
cell performance. Figure 9 shows the variation of the average output current density with the CFL
thickness (lCFL). Significant improvement is observed when lCFL is increased from 5 μm to 20 μm and
may be attributed to the increase of CFL active area as indicated in Figure 4. The output current
density is quite similar when lCFL is increased from 20 μm to 40 μm as neither the active CFL thickness
is extended nor the gas transport is affected appreciably. As shown in Figure 9, further increase of lCFL
may cause reduced output current density due to the increased concentration polarization. Therefore,
the optimal lCFL is between 20 μm and 40 μm.
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Figure 9. The variation of the average output current density with the CFL thickness.
3.3.4. Optimization of the AFL thickness
The effect of AFL thickness on the output current density is also studied and shown in Figure 10.
As shown in Figure 4, the active AFL thickness is only 4 μm. It is not surprising that the output current
density in Figure 10 decreases with lAFL.
Figure 10. The variation of the average output current density with the AFL thickness.
However, the decrease of the output current density is quite small and is less than 0.68% when lAFL
varies from 5 μm to 20 μm as the concentration polarization is not significant for such thin AFL layer.
An AFL thickness of less than 20 μm may be considered to be optimal.
4. Conclusions
A 2D finite element model for anode-supported planar SOFC stack cells with functional layers is
demonstrated. The model considers the microstructures of electrode material, the gas transfer process
in the porous electrode layers, the electronic conducting process in the electrode layers, the ionic
conducting process in the functional layers and electrolyte, the bulk electrochemical reactions in the
functional layers and the boundary electrochemical reactions at the functional layer/electrolyte
interfaces. The model was validated by comparison with experimental results on button cell I-V data.
Distributions of physical quantities in SOFC stack cells with functional layers are illustrated. The
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numerical model is also used to optimize geometry parameters of representative stack cells,
demonstrating its capabilities as a design and modeling tool for the development of SOFC technology.
Acknowledgements
We gratefully acknowledge the financial support of the Knowledge Innovation Program and the
Key Program of the Chinese Academy of Sciences (KJCX1.YW.07), the National High-tech R&D
Program of China (2007AA05Z156) and the National Science Foundation of China (10574114).
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